1) The document discusses five methods for generating orthogonal polynomials for discrete measures: Nowak's method, Fischer's method, Stieltjes method, modified Chebyshev method, and Lanczos method. 2) It compares the orthogonality, computational cost, and minimum polynomial order where the Stieltjes method starts to fail for these five methods. 3) It proposes an adaptive multi-element polynomial chaos method (ME-PCM) for stochastic partial differential equations driven by discrete random variables and demonstrates its accuracy on example problems.

1. Adaptive ME-PCM for SPDEs driven by discrete RVs !
Mengdi Zheng, Applied Mathematics, Brown University!
5 methods to generate orthogonal polynomials
for discrete measures:!
1. (Nowak’s method) S. Oladyshkin, W.
Nowak, Data-driven uncertainty quantification
using the arbitrary polynomial chaos
expansion, Reliability Engineering & System
Safety, 106 (2012), pp. 179–190. !
2. (Fischer’s method) H. J. Fischer, On
generating orthogonal polynomials for
discrete mea- sures, Z. Anal. Anwendungen,
17 (1998), pp. 183–205. !
3. (Stieltjes method) W. Gautschi, On
generating orthogonal polynomials, SIAM J.
Sci. Stat. Comp., 3 (1982), no.3, pp. 289–
317. !
4. (Modified Chebyshev method) the same
paper as above!
5. (Lanczos method) D. Boley, G. H. Golub, A
Comparing othogonality of 5 methods to
construct polynomials:
0 10 20 30 40 50 60 70 80 90 100
0
10
−5
10
−10
10
−15
10
−20
Comparing CPU time (cost) to construct the
polynomials by 5 methods:
0
10
−1
10
−2
10
−3
10
−4
Nowak
Stieltjes
Fischer
Modified Chebyshev
Lanczos
C*i2

2. n=100,p=1/2
10 20 40 80 100
10
polynomial order i
CPU time to evaluate orth(i)
Comparing the minimum polynomial orders that
the Stieltjes method starts to fail (Binomial):
160
140
120
100
80
60
40
20
0
0 20 40 60 80 100 120 140 160
n (p=1/10) for measure defined in (28)
polynomial order i
=1E−8
=1E−10
=1E−13
i = n

3. 10
polynomial order i
orth(i)
Nowak
Stieltjes
Fischer
Modified Chebyshev
Lanczos
n=100, p
=

4. 1
/2
Multi-element (ME) Gauss quadrature
integration theorem (new):
h-convergence of integration of GENZ1 function
over Binomial distributions (Lanczos/ME-PCM)
0
−2
10
−3
10
−4
10
−5
10
−6
10
1
10
10
N
es
absolute error
c=0.1,w=1
GENZ1
d=2
m=3
bino(120,1/2)
h-convergence of integration of GENZ4 function
over Binomial distributions (Lanczos/ME-PCM)
0
−9
10
−10
10
−11
10
−12
10
−13
10
1
10
10
N
es
absolute errors
c=0.1,w=1
GENZ4
d=2
m=3
bino(120,1/2)
Comparing sparse grid and tensor product grid in
8 dimensions by integration of GENZ1 function
over Binomial distribution (Lanczos/ME-PCM)
17 153 969 4845
−3
10
−4
10
−5
10
−6
10
−7
10
−8
10
−9
10
−10
10
r(k)
absolute error
sparse grid
tensor product grid
Genz1
sparse 8d
Bino(5,1/2)
1,...,8c
=0.1
1,...,8
w
1,...,8
=1
ADAPTIVE integration mesh of ME-PCM (idea)

7. Example: KdV equation/ homogeneous BC/
moment statistics!
Define errorr (for moment statistics)
ADAPTIVE V.s. NON-ADAPTIVE mesh
(moment statistics/KdV/Poisson RV/Nowak)!
−3
10
−4
10
−5
2 el, even grid
2 el, uneven grid
4 el, even grid
2 el, uneven grid
5 el, even grid
5 el, uneven grid

8. 4
2 3 4 5 6
−2
10
10
−3
10
−4
10
−5
Number of PCM points on each element
2 el, even grid
2 el, uneven grid
4 el, even grid
4 el, uneven grid
5 el, even grid
5 el, uneven grid

10. 2 3 4 5 6
10
Number of PCM points on each element
errors
errors
Details of this work please see: M. Zheng, X.
Wan, and G.E. Karniadakis, Adaptive-multi-element
polynomial chaos with discrete
measure: Algorithms and application to
SPDEs, Submitted to Applied Numerical
Mathematics, 2013.!
FA 9550-09-1-0613